Dust temperature in ALMA $\hbox{[C $\scriptstyle\rm II $]}$-detected high-$z$ galaxies

At redshift $z>5$ the far-infrared (FIR) continuum spectra of main-sequence galaxies are sparsely sampled, often with a single data point. The dust temperature $T_{\rm d, SED}$ thus has to be assumed in the FIR continuum fitting. This introduces large uncertainties regarding the derived dust mass ($M_{\rm d}$), FIR luminosity, and obscured fraction of the star formation rate. These are crucial quantities to quantify the effect of dust obscuration in high-$z$ galaxies. To overcome observations limitations, we introduce a new method that combines dust continuum information with the overlying \hbox{[C \scriptstyle\rm II ]} 158\mum line emission. By breaking the $M_{\rm d} - T_{\rm d, SED}$ degeneracy, with our method, we can reliably constrain the dust temperature with a single observation at $158\mu$m. This method can be applied to all ALMA and NOEMA \hbox{[C \scriptstyle\rm II ]} observations and exploited in ALMA Large Programs such as ALPINE and REBELS targeting \hbox{[C \scriptstyle\rm II ]} emitters at high-$z$. We also provide a physical interpretation of the empirical relation recently found between $molecular$ gas mass and \hbox{[C \scriptstyle\rm II ]} luminosity. We derive an analogous relation linking the $total$ gas surface density and \hbox{[C \scriptstyle\rm II ]} surface brightness. By combining the two, we predict the cosmic evolution of the surface density ratio $\Sigma_{\rm H_2} / \Sigma_{\rm gas}$. We find that $\Sigma_{\rm H_2} / \Sigma_{\rm gas}$ slowly increases with redshift, which is compatible with current observations at $0 < z < 4$.